On the complexity of computing the volume of a polyhedron
SIAM Journal on Computing
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
Sampling and integration of near log-concave functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
Simulated Annealing in Convex Bodies and an 0*(n4) Volume Algorithm
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Dispersion of Mass and the Complexity of Randomized Geometric Algorithms
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
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Volume computation is a traditional, extremely hard but highly demanding task. It has been widely studied and many interesting theoretical results are obtained in recent years. But very little attention is paid to put theory into use in practice. On the other hand, applications emerging in computer science and other fields require practically effective methods to compute/estimate volume. This paper presents a practical Monte Carlo sampling algorithm on volume computation/estimation and a corresponding prototype tool is implemented. Preliminary experimental results on lower dimensional instances show a good approximation of volume computation for both convex and non-convex cases. While there is no theoretical performance guarantee, the method itself even works for the case when there is only a membership oracle, which tells whether a point is inside the geometric body or not, and no description of the actual geometric body is given.